This subtopic studies Grothendieck groups and K_0, capturing additive invariants of exact or projective structures in algebra and geometry.

This subtopic studies Whitehead groups and K_1, focusing on automorphism-related invariants and their role in algebraic and geometric classification.

This subtopic studies Steinberg groups and related K_2 theory, emphasizing relations, symbols, and foundational constructions in algebraic K-theory.

This subtopic studies higher algebraic K-theory, extending low-dimensional K-groups to capture deeper structural and homotopical information.

This subtopic studies K-theory in geometry and topology, where vector bundles, characteristic classes, and topological invariants interact.

This subtopic studies K-theory in number theory and arithmetic, linking algebraic K-groups with arithmetic invariants and special values of L-functions.

This subtopic studies obstructions and regulators, focusing on invariants that detect when algebraic or geometric constructions fail to exist or can be measured analytically.

This subtopic studies topological K-theory, emphasizing vector bundles, Bott periodicity, and stable phenomena in topology.

This subtopic studies applications of K-theory to operator algebras, connecting C*-algebras, index theory, and noncommutative geometry.

This subtopic studies miscellaneous applications of K-theory, collecting additional contexts where K-theoretic ideas appear outside the core classifications.